On a class of q-Bernoulli and q-Euler polynomials

Nazim I Mahmudov

doi:10.1186/1687-1847-2013-108

Abstract

The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava-Pintér addition theorem is obtained. We give new identities involving q-Bernstein polynomials.

Highlights

1 Introduction Throughout this paper, we always make use of the following notation: N denotes the set of natural numbers, N denotes the set of nonnegative integers, R denotes the set of real numbers, C denotes the set of complex numbers
The q-polynomial coefficient is defined by n =
The second advantage is that we find the relation between q-Bernstein polynomials and Phillips q-Bernoulli polynomials and derive the formulas involving the q-Stirling numbers of the second kind, q-Bernoulli polynomials and Phillips q-Bernstein polynomials

Summary

Introduction

Throughout this paper, we always make use of the following notation: N denotes the set of natural numbers, N denotes the set of nonnegative integers, R denotes the set of real numbers, C denotes the set of complex numbers. The q-polynomial coefficient is defined by n =. Dqeq(z) = eq(z), DqEq(z) = Eq(qz), where Dq is defined by f (qz) – f (z) Dqf (z) := qz – z ,. We first give here the definitions of the q-Bernoulli and the q-Euler polynomials of higher order as follows. The q-Bernoulli numbers Bn(α,q) and polynomials Bn(α,q)(x, y) in x, y of order α are defined by means of the generating functions:. The q-Euler numbers En(α,q) and polynomials En(α,q)(x, y) in x, y of order α are defined by means of the generating functions: eq(t) +

It is obvious that

It is clear that